If the matrix \(W\)  is indefinite the PRESB preconditioner can not be directly applied to the real form of the system (REF ), but one can use some form of matrix splittings, similar to the classical alternating iteration method, see e.g. [1]}. Here the PRESB method can be used for each of the block matrices arising in the matrix splitting.
We present various forms of such methods with spectral analysis. The methods involve coupled inner-outer iterations. It is shown that the number of the outer iterations are insensitive to the inner solution accuracy, which can be utilized to lower the solution cost.
